A Linear Time Algorithm for Computing Max-Flow Vitality in Undirected Unweighted Planar Graphs

TL;DR

This paper introduces a linear time algorithm for computing the max-flow vitality of edges in unweighted planar graphs, extending efficient solutions from $st$-planar to all planar graphs.

Contribution

It presents the first optimal linear time algorithm for max-flow vitality in general unweighted planar graphs, broadening previous specialized solutions.

Findings

Algorithm runs in linear time for unweighted planar graphs.

Provides the first efficient solution for general planar graphs.

Extends max-flow vitality computation beyond $st$-planar graphs.

Abstract

The vitality of an edge in a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow value caused by the removal of that edge. The max-flow vitality problem has already been efficiently solved for $st$-planar graphs but has remained open for general planar graphs. For the first time our result provides an optimal solution for general planar graphs although restricted to the case of unweighted planar graphs.